A. A. Bolibruch, Yu. S. Osipov, and Ya. G. Sinai, editors

Every mathematician is aware of at least some of the great achievements of the mathematicians of the former Soviet Union. Many, I would guess, attach particular importance in their own work to the ideas of this or that Russian mathematician. Some are more aware than others of the way the Cold War worked to diminish contact between the Soviet Union and the community of mathematicians outside the communist countries, often leading to multiple discoveries of important results. Now for the first time we have a book documenting the life and achievements of that once vibrant mathematical culture, and of course it is impossible to review. Experts in any one area described here — there are 24 chapters (most of them ably translated by Roger Cooke), all of them technical to varying degrees — will have their own opinions on what is presented, and it is unlikely that anyone is an expert in all 24 fields. Those who know some of the people personally and were involved in some of the events described here will also have their judgements to make. I cannot stand under either heading, so it seems better to offer some more general remarks about what this book contains.

The structure of most chapters is that of a life and works, either of the author or of the group to which he belonged (all the authors are men). On a number of occasions what once were names become people, which is always pleasing. More importantly, what does emerge from many of these essays is the intense personal involvement of so many people in the life of mathematics. We can learn here that Aleksandrov regarded taking on a student as a commitment for life, one that could extend well beyond taking responsibility for their emergence as a research mathematician and include anything from standing up for them against the wishes of the bureaucracy to getting them tickets for concerts. We learn of a habit of seminars lasting for several hours, and long rolling discussions, as well as swimming trips and skiing trips (even with the blind but fearless Pontryagin). It would seem that the Soviet education system was rather good at spotting talent and fostering it, although we learn only indirectly here of anti-Semitism, apart from an article on the way Israel handled the migration of large numbers of Russian Jewish mathematicians.

A striking theme that comes up in a number of these essays is the way the balance was struck between pure and applied mathematics, and between mathematics and physics. This does not appear to have been greatly influenced by decisions from on high. At least, direct involvement of politicians is not on show here. Rather, it seems to have been the view of the culture, as it was for many years in France in the 19th century, that of course mathematics would stay close to physics, draw inspiration from it and make contributions to it. And to be sure, as the editors note, not everyone who was asked to contribute did so, and some who are not represented might have told a different story.

What the proper relationship between mathematics and physics should be was plainly a matter of some debate and disagreement. Thus in his essay Faddeev, who has considered himself a mathematical physicist for 40 years, regards mathematical physics as a subject that grew out of what was called theoretical physics 100 years ago, but became more mathematical with the books by Courant and Hilbert and Sobolev. Here, the emphasis was on partial differential equations and the calculus of variations. Then, in the 1960s, quantum mechanics added functional analysis to the mix, with spectral theory, Lie groups and their representations, and the Schrödinger operator. However, in Faddeev's own opinion, the principal aim of mathematical physics is 'to use mathematical intuition to derive genuinely new results in basic physics.' He predicts that as experiments in particle physics become ever more expensive, mathematical intuition will have an increasing role in guiding work in this area, and he cites work on Yang-Mills theory as a case in point.

In Sinai's essay, we learn that mathematicians took up physics in a serious way in the 1950s, led by Kolmogorov and Gel'fand, and that physicists started to travel in the opposite direction a little later. Inspired by Kolmogorov's work on the entropy of a dynamical system, Sinai showed that many systems have positive entropy, and this caught the attention of plasma physicists. In his view, 'mathematical physics is studied by mathematicians who obtain rigorous mathematical theorems and physicists who obtain rigorous results', and the mathematicians recognise that 'the results they are obtaining are not new from the point of view of a physicist.' However, mathematics has the advantage over physics that classical music has over pop: fewer fans, but it lasts through the ages.

Topology was one of the great strengths of the Russian school. Here among other essays is a moving recollection by Mischenko of Aleksandrov, Pontryagin, and their scientific schools. Mischenko argues, with quotes from Pontryagin's autobiography, that Pontryagin's switch from topology to control theory arose from a growing anxiety about what mathematics should be studied. In his view, the beauty of mathematics, being accessible only to a few, is not enough, and so most mathematicians 'should turn to the original sources in their work, that is, to the applications of mathematics.'

There is much more in this book than can be discussed here: the recurring influence of the Hilbert Problems (essays by Arnold, Bolibruch, Il'yashenko, Matiyasevich, Nesterenko), the sinuous overlap of different parts of mathematics, much more on mathematical physics and applied mathematics. As a result, other essays can at most be noted here, and I would like to mention the one by Vershik, who writes in elegiac tones on what he considers the rise and fall of functional analysis in the 20th Century, in an essay entitled 'The Life and Fate of Functional Analysis in the Twentieth Century', the title echoing that of Vassily Grossmann's great novel. The essay is followed by Vitushkin's essay 'Half a Century as one day' — this title echoes Chingiz Aitmatov's 'The Day lasts more than a hundred Years' — another essay equally interesting for the mathematics and the account of the ways mathematics was studied.

Several authors note that with the collapse of communism there was a great emigration of highly talented mathematicians, and the Russian mathematical schools sustained a great blow. There are signs that something is being done about this, but one must be sanguine. The work described here was largely done under a political regime with no real interest in mathematics but informed by enough of the beneficial aspects of the Enlightenment and socialist thought to leave it well alone. If anything, the world we all now inhabit is less willing to create mathematicians and let them be, and more demanding of immediate, profitable results. One way to look at this book is too see how much can be done with benign neglect and inspiration, and to draw hope and ideas from that.

Jeremy Gray is Professor of the History of Mathematics at the Centre for the History of the Mathematical Sciences of the Open University, in Milton Keynes, UK. He is the author of many books on the history of mathematics.